The results above show that, in the case where degradation and normalized synthesis rates are allowed to be different for each element of the network, it becomes more difficult for the system to sustain the coexpression of many elements. Indeed, equation 3 implies that the weakest element (in terms of the ratio of the maximal synthesis rate to the product of the degradation rate and the dissociation constant for heterodimer formation with class A proteins) that is "on" cannot be much weaker than the other ones which are being coexpressed (an intuitive result, since a weak element would be too easily repressed by the other ones, and wouldn't stay "on" in their presence). In addition to that, the competition level restricts the number of elements which can be coexpressed, in the same way as when degradation and normalized synthesis rates are all equal.
There is a great variety of ways in which a switch network can be led from a state of coexpression of all its elements to a state where only one is expressed, by changes in the competition level and in the synthesis and degradation rates. We show here two numerical simulations, to illustrate equation 3. In Figure 1, the competition level is increased, in a network in which elements have different normalized synthesis to degradation ratios; the weakest nonzero element is turned off every time the competition reaches a threshold. In Figure 2, the competition level is kept constant, but one element is made progressively stronger, and turns off all the other ones. Of course, the alteration of all parameters at the same time would be a plausible biological situation.


The networks studied here have been described in the context of class A and class B bHLH heterodimerization, but they could have a much wider relevance. Hox proteins, crucial determinants of tissue identity, have been shown to depend heavily on common binding partners of the PBC and Meis families (Mann, 1998). A subfamily of bHLHleucine zipper proteins shows tissuespecific expression, homo and heterodimerization, and alternative splicing of dominantnegative forms (Kuiper, 2004). Myc and Max, which have opposite roles on cell growth and proliferation, form homodimers and heterodimers with Mad, with different affinities (Grinberg, 2004).
Networks in which each element needs to repress all others can easily be created with competition for a common heterodimerization partner, rather than active repression of all other elements. Networks including other forms of crossrepression and asymmetrical topologies would also be of interest to study cellular differentiation, and are currently under investigation.